scholarly journals Twisted product CR-submanifolds in a locally conformal Kähler space forms

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1915-1925
Author(s):  
Vittoria Bonanzinga ◽  
Koji Matsumoto
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Twisted Product ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
Locally Conformal Kähler ◽  
The Mean ◽  
Locally Conformal ◽  
Cr Submanifolds

Certain twisted product CR-submanifolds in a K?hler manifold and some inequalities of the second fundamental form of these submanifolds are presented ([14]). Then the length of the second fundamental form of a twisted product CR-submanifold in a locally conformal K?hler manifold is considered (2013), ([15]). In this paper, we consider the relation of the mean curvature and the length of the second fundamental form in two twisted product CR-submanifolds in a locally conformal K?hler space forms.

scholarly journals Twisted product CR-submanifolds in a locally conformal Kaehler manifold

2013 ◽  
Vol 94 (108) ◽  
pp. 131-140
Author(s):  
Koji Matsumoto ◽  
Zerrin Şentürk
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Kaehler Manifold ◽  
Twisted Product ◽  
Equality Case ◽  
The Second Fundamental Form ◽  
Locally Conformal ◽  
Cr Submanifolds

Recently, we have researched certain twisted product CR-submanifolds in a Kaehler manifold and some inequalities of the second fundamental form of these submanifolds [11]. We consider here two kinds of twisted product CR-submanifolds (the first and the second kind) in a locally conformal Kaehler manifold. In these submanifolds, we give inequalities of the second fundamental form (see Theorems 5.1 and 5.2) and consider the equality case of these.

A note on rigidity of spacelike self-shrinkers

2020 ◽  
Vol 31 (05) ◽  
pp. 2050035
Author(s):  
Yong Luo ◽  
Hongbing Qiu
Keyword(s):  
Euclidean Space ◽  
Growth Condition ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Integral Method ◽  
Acta Math ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
The Mean ◽  
A Minor

By using the integral method, we prove a rigidity theorem for spacelike self-shrinkers in pseudo-Euclidean space under a minor growth condition in terms of the mean curvature and the second fundamental form, which generalizes Theorem 1.1 in [H. Q. Liu and Y. L. Xin, Some Results on Space-Like Self-Shrinkers, Acta Math. Sin. (Engl. Ser.) 32(1) (2016) 69–82].

scholarly journals Self-Expanders of the Mean Curvature Flow

2021 ◽  
Author(s):  
Knut Smoczyk
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Vector ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
The Mean ◽  
Mean Convex

AbstractWe study self-expanding solutions $M^{m}\subset \mathbb {R}^{n}$ M m ⊂ ℝ n of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function |A|2/|H|2 attains a local maximum, where A denotes the second fundamental form and H the mean curvature vector of M. If the principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ|2/|H|2, where Aξ is the second fundamental form with respect to ξ. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex self-expander that is asymptotic to a cone must be strictly convex.

Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions

2009 ◽  
Vol 61 (3) ◽  
pp. 641-655
Author(s):  
Sadahiro Maeda ◽  
Seiichi Udagawa
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Space Form ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Isometric Immersions ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
The Mean

Abstract.For an isotropic submanifold Mn (n ≧ 3) of a space form of constant sectional curvature c, we show that if the mean curvature vector of Mn is parallel and the sectional curvature K of Mn satisfies some inequality, then the second fundamental form of Mn in is parallel and our manifold Mn is a space form.

scholarly journals A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1238
Author(s):  
Pablo Alegre ◽  
Joaquín Barrera ◽  
Alfonso Carriazo
Keyword(s):  
Closed Form ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Space Forms ◽  
Equality Case ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Sasakian Space Forms ◽  
Maslov Form ◽  
The Mean

The Maslov form is a closed form for a Lagrangian submanifold of C m , and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal.

scholarly journals Space-like submanifolds with parallel mean curvature in de Sitter spaces

2000 ◽  
Vol 69 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Chongzhen Ouyang ◽  
Zhenqi Li
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Parallel Mean Curvature Vector ◽  
Complete Space ◽  
The Second Fundamental Form ◽  
Parallel Mean Curvature

AbstractThis paper investigates complete space-like submainfold with parallel mean curvature vector in the de Sitter space. Some pinching theorems on square of the norm of the second fundamental form are given

scholarly journals SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPHERES

2011 ◽  
Vol 54 (1) ◽  
pp. 67-75 ◽  
Author(s):  
QIN ZHANG
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Unit Sphere ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
Closed Hypersurface ◽  
Small Constant

AbstractLet Mn be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1(1), n ≤ 8 and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.

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